#10409681

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4489 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2)(3,5)(4,6)(7,8)(9,10)(11,19)(12,16)(13,14)(15,18)(17,24)(20,23)(21,22)(25,26)(27,32)(28,30)(29,36)(31,35)(33,37)(34,38)(39,40);(2,3)(6,7)(25,27)(26,31)(28,32)(29,34)(30,35)(33,38)(36,39)(37,40);(9,11,15,12)(10,16,18,19)(13,20,17,22)(14,21,24,23)(25,26,30,28)(27,32,35,31)(29,33,37,36)(34,39,40,38);(2,3)(4,8)(9,12,15,11)(10,16,18,19)(13,20,17,22)(14,23,24,21)(25,28,30,26)(27,32,35,31)(29,36,37,33)(34,39,40,38);(1,3,5,2)(4,7,8,6)(9,13,15,17)(10,14,18,24)(11,20,12,22)(16,21,19,23)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,38)(37,40);(1,4,5,8)(2,6,3,7)(9,14)(10,17)(11,21)(12,23)(13,18)(15,24)(16,22)(19,20)(25,29,30,37)(26,33,28,36)(27,34,35,40)(31,38,32,39)

Polycyclic presentation:

2^9|c^2=h,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10316877);